In the number series 2, 3, 7, 16, 32, ?, which number should replace the question mark so that the pattern is preserved?

Introduction / Context:
This question is based on a number series where the difference between consecutive terms is not constant but follows a recognisable pattern. Identifying the correct pattern in these differences is crucial to determining the missing number and is a common type of challenge in quantitative reasoning sections.

Given Data / Assumptions:
- The series is: 2, 3, 7, 16, 32, ?- All terms are positive and increasing.- The pattern appears to involve changing differences rather than simple multiplication.

Concept / Approach:
To solve such series, first compute the differences between consecutive terms. If the differences themselves form a structured pattern, such as squares, cubes, or another arithmetic sequence, you can extend that pattern to find the next difference and consequently the next term in the original series.

Step-by-Step Solution:
- Calculate differences: 3 - 2 = 1, 7 - 3 = 4, 16 - 7 = 9, 32 - 16 = 16.- The difference sequence is: 1, 4, 9, 16.- These are perfect squares: 1^2, 2^2, 3^2, 4^2.- Naturally, the next difference in this pattern should be 5^2 = 25.- Add this next difference to the last known term: 32 + 25 = 57.- Therefore, the missing term in the series is 57.

Verification / Alternative check:
- Rebuild the series using the rule: start from 2 and add 1^2, 2^2, 3^2, 4^2, 5^2.- 2 + 1 = 3, 3 + 4 = 7, 7 + 9 = 16, 16 + 16 = 32, 32 + 25 = 57.- Every step matches the given series and yields 57 as the next term, confirming the logic.

Why Other Options Are Wrong:
- 49, 52, and 54 do not satisfy the pattern of adding consecutive square numbers.- Using any of these alternatives would break the neat square-difference sequence and result in inconsistent future terms.- Only 57 can be expressed as 32 plus 5^2 in line with the established rule.

Common Pitfalls:
One common mistake is to look for multiplication or simple constant differences and give up when none is found. Another pitfall is noticing that the differences increase but not recognising that they are perfect squares. When the series grows steadily but not explosively, always check whether the differences themselves form a classic sequence like squares or cubes.

Final Answer:
The number that correctly completes the series is 57, so the correct option is 57.